Define \(a_n = s_0 \log n/n^2\) . We will show that \(\int _{-a_n}^{a_n}p_{HS}(\eta ; \tau )\, d\eta \ge 1 - n^{-b}\)  for some \(b > 0\)  under the assumptions outlined in Theorem REF . The Horseshoe prior density does not assume a closed form; however, [1]} showed that the density admits tight lower and upper bounds given by,
\(\dfrac{1}{\tau (2\pi )^{3/2}}\log \left( 1 + \dfrac{4\tau ^2}{\theta ^2}\right) < p_{HS}(\theta \mid \tau ) <\dfrac{2}{\tau (2\pi )^{3/2}}\log \left( 1 + \dfrac{2\tau ^2}{\theta ^2}\right).\) 
